The above observation has a convenient reformulation in terms of category theory. Let 𝖱𝖬𝗈𝖽 denote the category of R-modules, and 𝖲𝖾𝗍 the category of sets. Consider the adjoint functorsU:𝖱𝖬𝗈𝖽→𝖲𝖾𝗍, the forgetful functor that maps an R-module to its underlying set, and F:𝖲𝖾𝗍→𝖱𝖬𝗈𝖽,
the free modulefunctor that maps a set to the free R-module generated by that set. To say that U is right-adjoint to F is the same as saying that every set map from B to U⁢(N), the set underlying N, corresponds naturally and bijectively to an R-module homomorphism from M=F⁢(B) to N.

Similarly, given a map f2:B2→N, we may define the
bilinear extension

φ2:

M2→N

(m,n)

↦∑b∈B∑c∈Bmb⁢nc⁢f2⁢(b,c),

which is the unique bilinear map from M2 to N whose restriction
to B2 is f2.

Generally, for any positive integer n and a map fn:Bn→N, we may define the n-linear extension

Usage

The notion of linear extension is typically used as a
manner-of-speaking. Thus, when a multilinear map is defined
explicitly in a mathematical text, the images of the basis elements
are given accompanied by the phrase “by multilinear extension” or
similar.